Unit

Let $𝑅$ be a ring, and $𝑎 \in 𝑅$ . Then

$𝑎$ is a left unit iff $𝑎 𝑏 = 1$ for some $𝑏$ ;
$𝑎$ is a right unit iff $𝑏 𝑎 = 1$ for some $𝑏$ ;
$𝑎$ is a unit iff it is both a left unit and right unit.

By the usual argument, the inverse of an ambidextrous unit is unique, and these form the group of units. A ring in which every nonzero element is a unit is called a Division ring.

As morphisms

Let $𝑅 ――$ denote the multiplicative monoid of a ring $𝑅$ viewed as a category. Then $𝑎 \in 𝑅$ is

a left unit iff it is split epic;
a right unit iff it is split monic;
a unit iff it is an isomorphism.

If we view $Λ (𝑎)$ and $P (𝑎)$ as functions on $𝑅$ , then $𝑎 \in 𝑅$ is¹

a left unit iff $Λ (𝑎)$ is surjective iff $P (𝑎)$ is injective iff $𝑎$ is not a right zero-divisor;
a right unit iff $P (𝑎)$ is surjective iff $Λ (𝑎)$ is surjective iff $𝑎$ is not a left zero-divisor;

Unit

As morphisms

See also

Footnotes